1985
DOI: 10.1007/bf01389877
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Is Gauss quadrature optimal for analytic functions?

Abstract: Summary. We consider the problem of optimal quadratures for integrands f: [-1,1]--,~ which have an analytic extension f to an open disk D, of radius r about the origin such that Ill_-< 1 on D,. If r= 1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an "infinite" penalty for using Gauss quadrature. On the other hand, if r>l, Gauss quadrature is almost optimal. These results hold … Show more

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Cited by 19 publications
(10 citation statements)
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“…On the other hand, if the integrands are flatter, i.e., γ is small, with respect to the variance of the Gaussian integration weight α, then the Gauss-Hermite quadratures will work very well. Note that a similar property of the Gauss quadratures also holds for univariate integration over finite intervals as demonstrated in e.g., [10] and [12].…”
Section: Introductionmentioning
confidence: 87%
“…On the other hand, if the integrands are flatter, i.e., γ is small, with respect to the variance of the Gaussian integration weight α, then the Gauss-Hermite quadratures will work very well. Note that a similar property of the Gauss quadratures also holds for univariate integration over finite intervals as demonstrated in e.g., [10] and [12].…”
Section: Introductionmentioning
confidence: 87%
“…where |ε n | ≤ 3. Proposition 5 applied top n =p n 2 −t then provides the bound (13). Turning to Algorithm 1, let p 0 , q 0 , d 0 denote the values of the variables p, q, den before the loop, and p k , q k , d k their values at the end of iteration k. Consider the sequencep k = 2 −t q k−1 /d k−1 , k ≥ 1, extended byp 0 = 1 and an arbitrary (finite)p −1 .…”
Section: Algorithm 1 Evaluation Of Legendre Polynomials In Gmp Fixed-mentioning
confidence: 96%
“…The sequence (p n ) n≥0 defined by (10) satisfies (13) |p n 2 −t − p n | ≤ 0.75 (n + 1)(n + 1) 2 −t , n ≥ 0.…”
Section: Algorithm 1 Evaluation Of Legendre Polynomials In Gmp Fixed-mentioning
confidence: 99%
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